SOLUTION: Find the greatest value of "xy". If ( (a^2 x^4) + ( b^2 y^4) ) = c^6. ?

Algebra ->  Inequalities -> SOLUTION: Find the greatest value of "xy". If ( (a^2 x^4) + ( b^2 y^4) ) = c^6. ?      Log On


   

Question 955351: Find the greatest value of "xy".
If ( (a^2 x^4) + ( b^2 y^4) ) = c^6. ?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
a%5E2+x%5E4+%2B+b%5E2+y%5E4+=+c%5E6
b%5E2y%5E4=c%5E6-a%5E2x%5E4
y%5E4=%28c%5E6-a%5E2x%5E4%29%2Fb%5E2
y=%28%28c%5E6-a%5E2x%5E4%29%2Fb%5E2%29%5E%281%2F4%29
So then,
Z=xy
Z=x%28%28c%5E6-a%5E2x%5E4%29%2Fb%5E2%29%5E%281%2F4%29
Taking the derivative of Z with respect to x,

To find the extrema, set the derivative equal to zero.
c%5E6-2a%5E2x%5E4=0
2a%5E2x%5E4=c%5E6
x%5E4=c%5E6%2F%282a%5E2%29
So then,
a%5E2%28c%5E6%2F%282a%5E2%29%29%2Bb%5E2y%5E4=c%5E6
c%5E6%2F2%2Bb%5E2y%5E4=c%5E6
b%5E2y%5E4=c%5E6%2F2}
y%5E4=c%5E6%2F%282b%5E2%29
So then,
x%5E4y%5E4=%28c%5E6%2F%282a%5E2%29%29%28c%5E6%2F%282b%5E2%29%29
%28xy%29%5E4=c%5E12%2F%284a%5E2b%5E2%29
and finally,
xy%5Bmax%5D=%28c%5E12%5E%281%2F4%29%29%2F%284a%5E2b%5E2%29%5E%281%2F4%29
highlight%28xy%5Bmax%5D=c%5E3%2Fsqrt%282ab%29%29